The function must accept as arguments two elements of sequence or the results from combining those elements. The function must also be able to accept no arguments.

If key is supplied, it is used is used to extract the values to reduce. The key function is applied exactly once to each element of sequence in the order implied by the reduction order but not to the value of initial-value, if supplied. The key function typically returns part of the element of sequence. If key is not supplied or is nil, the sequenceelement itself is used.

The reduction is left-associative, unless from-end is true in which case it is right-associative.

If initial-value is supplied, it is logically placed before the subsequence (or after it if from-end is true) and included in the reduction operation.

In the normal case, the result of reduce is the combined result of function's being applied to successive pairs of elements of sequence. If the subsequence contains exactly one element and no initial-value is given, then that element is returned and function is not called. If the subsequence is empty and an initial-value is given, then the initial-value is returned and function is not called. If the subsequence is empty and no initial-value is given, then the function is called with zero arguments, and reduce returns whatever function does. This is the only case where the function is called with other than two arguments.